Counterexample. Identify the points of the Fano plane with the numbers $1,2,3,4,5,6,7$ and call the lines $\ell_1,\ell_2,\ell_3,\ell_4,\ell_5,\ell_6,\ell_7.$

Let $N=\{0,1,2,3,4,5,6,7\}.$ Let $P$ contain the $5$-element sets $x_i=N\setminus\ell_i$ and the $4$-element sets $y_i=\ell_i\cup\{0\}$ where $i=1,\dots,7.$ Conditions (1)–(4) are clearly satisfied.

Let $\alpha=\frac17\sum_{i=1}^71_{x_i}$ and $\beta=\frac17\sum_{i=1}^71_{y_i}.$

Then $\alpha_0=\beta_0=1$ while, for $1\le i\le7,$ we have $\alpha_i=\frac47$ and $\beta_i=\frac37.$

Counterexample.

$N=\{0,1,2,3,4,5,6\}$

$P=\{u,v,w,y,z\}$

$u=\{0,1,3,5\},\ v=\{0,2,4,6\},\ x=\{0,1,2\},\ y=\{0,3,4\},\ z=\{0,5,6\}$

$\alpha=\frac12(1_u+1_v),\ \beta=\frac13(1_x+1_y+1_z)$